Rejoinder: ‘Multi-scale Fisher’s independence test for multivariate dependence’

Abstract

1. Introduction We wish to start by expressing our gratitude to the Editorial Board for giving us the opportunity to discuss our article, and to all of the discussants for their very insightful, thought-provoking contributions. These discussions provide a more comprehensive review of the vast relevant literature along with more in-depth views on several popular approaches for multivariate dependency testing, including computationally efficient approximation to kernel methods ( Schrab et al., 2022), the k -nearest-neighbour-based mutual information test ( Berrett, 2022) and the binary-expansion-test-based framework ( Lee et al., 2022a). The discussions also complement our paper with additional numerical experiments that shed light on the statistical and computational strengths and weaknesses of the various approaches, and suggest interesting future research directions such as testing functional dependency ( Lee et al., 2022a). We focus our response on three important practical considerations raised in the discussions: (i) the ability to characterize the dependency structure, (ii) the statistical power of the multi-scale Fisher’s independence test, Multi FIT, on various dependencies and (iii) the connection between Multi FIT and other methods.